课程名称︰工程数学 课程性质︰大二必修 课程教师︰施文彬 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰2008.06.16 考试时限（分钟）：110 mins 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Rule: No calculators are allowed. You are allowed to bring an A4 size information sheet. Points will not be given without providing details of your calculation. Please carry out all integrations in your calculation. Good luck ! 1.(25%) e^(6iz)sin(2z) Given f(z) = ─────── (1+iz)^2 (a) Find u and v so that f(z) = u(x,y)+iv(x,y). (b) Determine all points at which Cauchy-Riemann equations are satisfied ,and determine all points at which f(z) is differentiable. (c) Determine all residues of f(z). (d) Apply Residue Theorem to find the inverse Fourier transform of the e^(6iω)sin(2ω) function ───────── (1+iω)^2 2.(20%) d^2y d^2y Solve the wave equation ── = c^2(──) on the line for c = 2 and the given dt^2 dx^2 initial conditions: dy ╭ cos(πx) , -1.5≦x≦1.5 y(x,0) = xe^(-|x|) and ─(x,0) = ┤ dt ╰ 0 , |x|>1.5 3.(20%) (a) For what value(s) of "L" does the boundary value problem , y" + 16y = 0 , y(0) = 0 , y(L) = 0 , L>0 have a nontrivial solution ? (b) Write down the solution corresponding to the value(s) of "L" found in part(a). (c) For what values of "L" does the boundary value problem have a unique solution ? 4.(20%) (a) Solve (y'/x^3)' + (1+35/9x^2)y = 0 by letting u = y/x^2. (b) Show that for any real number v , x^2Jv"(x) = (v^2-v-x^2)Jv(x) + xJv+1(x). 5.(15%) d^2u du Consider the one dimensional heat transfer problem ── = 9─ , 0≦x≦5 dx^2 dt with boundary conditions u(0,t) = 0 ,u(5,t) = 4 ,t>0 and initial conditions u(x,0) = 0 , 0≦x≦5. (a) Find the long time , i.e. time independent , solution reached as t→∞. (b) Find the time-dependent solution u(x,t) that satisfies the given boundary and initial conditions. --

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